Evaluate Using the Given Value x^2-5y^2+412x^2+y^2=17

$x^{2} - 5 y^{2} + 412 x^{2} + y^{2} = 17$

Step 1

Simplify $x^{2} - 5 y^{2} + 412 x^{2} + y^{2}$ .

Add $x^{2}$ and $412 x^{2}$ .

$413 x^{2} - 5 y^{2} + y^{2} = 17$

Add $- 5 y^{2}$ and $y^{2}$ .

$413 x^{2} - 4 y^{2} = 17$

Step 2

Add $4 y^{2}$ to both sides of the equation.

$413 x^{2} = 17 + 4 y^{2}$

Step 3

Divide each term in $413 x^{2} = 17 + 4 y^{2}$ by $413$ and simplify.

Divide each term in $413 x^{2} = 17 + 4 y^{2}$ by $413$ .

$\frac{413 x^{2}}{413} = \frac{17}{413} + \frac{4 y^{2}}{413}$

Simplify the left side.

Cancel the common factor of $413$ .

Cancel the common factor.

$\frac{413 x^{2}}{413} = \frac{17}{413} + \frac{4 y^{2}}{413}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{17}{413} + \frac{4 y^{2}}{413}$

Step 4

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{17}{413} + \frac{4 y^{2}}{413}}$

Step 5

Simplify $\pm \sqrt{\frac{17}{413} + \frac{4 y^{2}}{413}}$ .

Combine the numerators over the common denominator.

$x = \pm \sqrt{\frac{17 + 4 y^{2}}{413}}$

Rewrite $\sqrt{\frac{17 + 4 y^{2}}{413}}$ as $\frac{\sqrt{17 + 4 y^{2}}}{\sqrt{413}}$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}}}{\sqrt{413}}$

Multiply $\frac{\sqrt{17 + 4 y^{2}}}{\sqrt{413}}$ by $\frac{\sqrt{413}}{\sqrt{413}}$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}}}{\sqrt{413}} \cdot \frac{\sqrt{413}}{\sqrt{413}}$

Combine and simplify the denominator.

Multiply $\frac{\sqrt{17 + 4 y^{2}}}{\sqrt{413}}$ by $\frac{\sqrt{413}}{\sqrt{413}}$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{\sqrt{413} \sqrt{413}}$

Raise $\sqrt{413}$ to the power of $1$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{{\sqrt{413}}^{1} \sqrt{413}}$

Raise $\sqrt{413}$ to the power of $1$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{{\sqrt{413}}^{1} {\sqrt{413}}^{1}}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{{\sqrt{413}}^{1 + 1}}$

Add $1$ and $1$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{{\sqrt{413}}^{2}}$

Rewrite ${\sqrt{413}}^{2}$ as $413$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{413}$ as $413^{\frac{1}{2}}$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{{(413^{\frac{1}{2}})}^{2}}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{413^{\frac{1}{2} \cdot 2}}$

Combine $\frac{1}{2}$ and $2$ .

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{413^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{413^{\frac{2}{2}}}$

Rewrite the expression.

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{413^{1}}$

Evaluate the exponent.

$x = \pm \frac{\sqrt{17 + 4 y^{2}} \sqrt{413}}{413}$

Combine using the product rule for radicals.

$x = \pm \frac{\sqrt{(17 + 4 y^{2}) \cdot 413}}{413}$

Reorder factors in $\pm \frac{\sqrt{(17 + 4 y^{2}) \cdot 413}}{413}$ .

$x = \pm \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

Step 6

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

$x = - \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

$x = \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

$x = - \frac{\sqrt{413 (17 + 4 y^{2})}}{413}$

Evaluate Using the Given Value x^2-5y^2+412x^2+y^2=17

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